Schrödinger group

Schrödinger algebra

The Schrödinger algebra is the Lie algebra of the Schrödinger group. It is not semi-simple. In one space dimension, it can be obtained as a semi-direct sum of the Lie algebra sl(2,R) and the Heisenberg algebra; similar constructions apply to higher spatial dimensions.

It contains a Galilei algebra with central extension.

${\displaystyle [J_a,J_b]=i{ϵ}_{abc}{J}_c,}$

${\displaystyle [J_a,P_b]=i{ϵ}_{abc}{P}_c,}$

${\displaystyle [J_a,K_b]=i{ϵ}_{abc}{K}_c,}$

${\displaystyle [P_a,P_b]=0,[K_a,K_b]=0,[K_a,P_b]=i{\delta }_{ab}M,}$

${\displaystyle [H,J_a]=0,[H,P_a]=0,[H,K_a]=i{P}_a.}$

where ${\displaystyle J_a,P_a,K_a,H}$ are generators of rotations (angular momentum operator), spatial translations (momentum operator), Galilean boosts and time translation (Hamiltonian) respectively. (Notes: ${\displaystyle i}$ is the imaginary unit, ${\displaystyle i^2=-1}$. The specific form of the commutators of the generators of rotation ${\displaystyle J_a}$ is the one of three-dimensional space, then ${\displaystyle a,b,c=1,\dots ,3}$.). The central extension M has an interpretation as non-relativistic mass and corresponds to the symmetry of Schrödinger equation under phase transformation (and to the conservation of probability).

There are two more generators which we shall denote by D and C. They have the following commutation relations:

${\displaystyle [H,C]=iD,[C,D]=-2iC,[H,D]=2iH,}$

${\displaystyle [P_a,D]=i{P}_a,[K_i,D]=-i{K}_a,}$

${\displaystyle [P_a,C]=-i{K}_a,[K_a,C]=0,}$

${\displaystyle [J_a,C]=[J_a,D]=0.}$

The generators H, C and D form the sl(2,R) algebra.

A more systematic notation allows to cast these generators into the four (infinite) families ${\displaystyle X_n,Y_m^{(j)},M_n}$ and ${\displaystyle R_n^{(jk)}=-R_n^{(kj)}}$, where n ∈ ℤ is an integer and m ∈ ℤ+1/2 is a half-integer and j,k=1,...,d label the spatial direction, in d spatial dimensions. The non-vanishing commutators of the Schrödinger algebra become (euclidean form)

${\displaystyle [X_n,X_{n^{\prime }}]=(n-n^{\prime })X_{n+n^{\prime }}}$

${\displaystyle [X_n,Y_m^{(j)}]=\left(\frac{n}{2}-m\right)Y_{n+m}^{(j)}}$

${\displaystyle [X_n,M_{n^{\prime }}]=-n^{\prime }{M}_{n+n^{\prime }}}$

${\displaystyle [X_n,R_{n^{\prime }}^{(jk)}]=-n^{\prime }{R}_{n^{\prime }}^{(jk)}}$

${\displaystyle [Y_m^{(j)},Y_{m^{\prime }}^{(k)}]={\delta }_{j,k}(m-m^{\prime })M_{m+m^{\prime }}}$

${\displaystyle [R_n^{(ij)},Y_m^{(k)}]={\delta }_{i,k}{Y}_{n+m}^{(j)}-{\delta }_{j,k}{Y}_{n+m}^{(i)}}$

${\displaystyle [R_n^{(ij)},R_{n^{\prime }}^{(kl)}]={\delta }_{i,k}{R}_{n+n^{\prime }}^{(jl)}+{\delta }_{j,l}{R}_{n+n^{\prime }}^{(ik)}-{\delta }_{i,l}{R}_{n+n^{\prime }}^{(jk)}-{\delta }_{j,k}{R}_{n+n^{\prime }}^{(il)}}$

The Schrödinger algebra is finite-dimensional and contains the generators ${\displaystyle X_{-1,0,1},Y_{-1/2,1/2}^{(j)},M_0,R_0^{(jk)}}$. In particular, the three generators ${\displaystyle X_{-1}=H,X_0=D,X_1=C}$ span the sl(2,R) sub-algebra. Space-translations are generated by ${\displaystyle Y_{-1/2}^{(j)}}$ and the Galilei-transformations by ${\displaystyle Y_{1/2}^{(j)}}$.

In the chosen notation, one clearly sees that an infinite-dimensional extension exists, which is called the Schrödinger–Virasoro algebra. Then, the generators ${\displaystyle X_n}$ with n integer span a loop-Virasoro algebra. An explicit representation as time-space transformations is given by, with n ∈ ℤ and m ∈ ℤ+1/2

${\displaystyle X_n=-t^{n+1}{\mathrm{\partial }}_t-\frac{n+1}{2}{t}^n\overrightarrow{r}\cdot {\mathrm{\partial }}_{\overrightarrow{r}}-\frac{n(n+1)}{4}\mathcal{M}{t}^{n-1}\overrightarrow{r}\cdot \overrightarrow{r}-\frac{x}{2}(n+1)t^n}$

${\displaystyle Y_m^{(j)}=-t^{m+1/2}{\mathrm{\partial }}_{r_j}-\left(m+\frac{1}{2}\right)\mathcal{M}{t}^{m-1/2}{r}_j}$

${\displaystyle M_n=-t^n\mathcal{M}}$

${\displaystyle R_n^{(jk)}=-t^n\left(r_j{\mathrm{\partial }}_{r_k}-r_k{\mathrm{\partial }}_{r_j}\right)}$

This shows how the central extension ${\displaystyle M_0}$ of the non-semi-simple and finite-dimensional Schrödinger algebra becomes a component of an infinite family in the Schrödinger–Virasoro algebra. In addition, and in analogy with either the Virasoro algebra or the Kac–Moody algebra, further central extensions are possible. However, a non-vanishing result only exists for the commutator ${\displaystyle [X_n,X_{n^{\prime }}]}$, where it must be of the familiar Virasoro form, namely

${\displaystyle [X_n,X_{n^{\prime }}]=(n-n^{\prime })X_{n+n^{\prime }}+\frac{c}{12}(n^3-n){\delta }_{n+n^{\prime },0}}$

or for the commutator between the rotations ${\displaystyle R_n^{(jk)}}$, where it must have a Kac-Moody form. Any other possible central extension can be absorbed into the Lie algebra generators.